Curvature and the Osculating Circle

The unit tangent and the turning it does

Formulas that avoid the arclength parameter

The principal normal points to the center of the bend

Acceleration splits into steering and pedaling

A circle of radius $5$ is traversed counterclockwise. What is its curvature?

For the helix $\mathbf{r}(t) = (\cos t, \sin t, t)$, what is $\kappa$?

For the parabola $y = x^2$, where is the curvature largest?

For the parabola $y = x^2$ at the point $(1, 1)$, the curvature is $\kappa = \dfrac{2}{(1+4)^{3/2}} = \dfrac{2}{5\sqrt{5}}$. Give the radius of curvature, the reciprocal, in the form a*sqrt(b)/c with integers.

A car maintains a constant speed of $20$ on a circular track of radius $50$. Give the magnitude of its acceleration, as a plain number or fraction.